Theorem cfss 10225

Description: There is a cofinal subset of 𝐴 of cardinality (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
cfss.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cfss	⊢ (Lim 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem cfss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfss.1	. . . . . 6 ⊢ 𝐴 ∈ V
2	1	cflim3 10222	. . . . 5 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
3		fvex 6874	. . . . . . 7 ⊢ (card‘𝑥) ∈ V
4	3	dfiin2 5001	. . . . . 6 ⊢ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
5		cardon 9904	. . . . . . . . . 10 ⊢ (card‘𝑥) ∈ On
6		eleq1 2817	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝑥) → (𝑦 ∈ On ↔ (card‘𝑥) ∈ On))
7	5, 6	mpbiri 258	. . . . . . . . 9 ⊢ (𝑦 = (card‘𝑥) → 𝑦 ∈ On)
8	7	rexlimivw 3131	. . . . . . . 8 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) → 𝑦 ∈ On)
9	8	abssi 4036	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On
10		limuni 6397	. . . . . . . . . . . 12 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
11	10	eqcomd 2736	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → ∪ 𝐴 = 𝐴)
12		fveq2 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
13	12	eqcomd 2736	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (card‘𝐴) = (card‘𝑥))
14	13	biantrud 531	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (∪ 𝐴 = 𝐴 ↔ (∪ 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥))))
15		unieq 4885	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
16	15	eqeq1d 2732	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 = 𝐴 ↔ ∪ 𝐴 = 𝐴))
17	1	pwid 4588	. . . . . . . . . . . . . . . . 17 ⊢ 𝐴 ∈ 𝒫 𝐴
18		eleq1 2817	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐴 ↔ 𝐴 ∈ 𝒫 𝐴))
19	17, 18	mpbiri 258	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → 𝑥 ∈ 𝒫 𝐴)
20	19	biantrurd 532	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴)))
21	16, 20	bitr3d 281	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (∪ 𝐴 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴)))
22	21	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ((∪ 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥))))
23	14, 22	bitr2d 280	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)) ↔ ∪ 𝐴 = 𝐴))
24	1, 23	spcev 3575	. . . . . . . . . . 11 ⊢ (∪ 𝐴 = 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
25	11, 24	syl 17	. . . . . . . . . 10 ⊢ (Lim 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
26		df-rex 3055	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)))
27		rabid 3430	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
28	27	anbi1i 624	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
29	28	exbii 1848	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
30	26, 29	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
31	25, 30	sylibr 234	. . . . . . . . 9 ⊢ (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥))
32		fvex 6874	. . . . . . . . . 10 ⊢ (card‘𝐴) ∈ V
33		eqeq1 2734	. . . . . . . . . . 11 ⊢ (𝑦 = (card‘𝐴) → (𝑦 = (card‘𝑥) ↔ (card‘𝐴) = (card‘𝑥)))
34	33	rexbidv 3158	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥)))
35	32, 34	spcev 3575	. . . . . . . . 9 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) → ∃𝑦∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥))
36	31, 35	syl 17	. . . . . . . 8 ⊢ (Lim 𝐴 → ∃𝑦∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥))
37		abn0 4351	. . . . . . . 8 ⊢ ({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅ ↔ ∃𝑦∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥))
38	36, 37	sylibr 234	. . . . . . 7 ⊢ (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅)
39		onint 7769	. . . . . . 7 ⊢ (({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅) → ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
40	9, 38, 39	sylancr 587	. . . . . 6 ⊢ (Lim 𝐴 → ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
41	4, 40	eqeltrid 2833	. . . . 5 ⊢ (Lim 𝐴 → ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
42	2, 41	eqeltrd 2829	. . . 4 ⊢ (Lim 𝐴 → (cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
43		fvex 6874	. . . . 5 ⊢ (cf‘𝐴) ∈ V
44		eqeq1 2734	. . . . . 6 ⊢ (𝑦 = (cf‘𝐴) → (𝑦 = (card‘𝑥) ↔ (cf‘𝐴) = (card‘𝑥)))
45	44	rexbidv 3158	. . . . 5 ⊢ (𝑦 = (cf‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥)))
46	43, 45	elab 3649	. . . 4 ⊢ ((cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
47	42, 46	sylib 218	. . 3 ⊢ (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
48		df-rex 3055	. . 3 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
49	47, 48	sylib 218	. 2 ⊢ (Lim 𝐴 → ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
50		simprl 770	. . . . . . . 8 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴})
51	50, 27	sylib 218	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
52	51	simpld 494	. . . . . 6 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ 𝒫 𝐴)
53	52	elpwid 4575	. . . . 5 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ⊆ 𝐴)
54		simpl 482	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → Lim 𝐴)
55		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
56		limord 6396	. . . . . . . . . . . 12 ⊢ (Lim 𝐴 → Ord 𝐴)
57		ordsson 7762	. . . . . . . . . . . 12 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
58	56, 57	syl 17	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
59		sstr 3958	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑥 ⊆ On)
60	58, 59	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ Lim 𝐴) → 𝑥 ⊆ On)
61		onssnum 10000	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑥 ⊆ On) → 𝑥 ∈ dom card)
62	55, 60, 61	sylancr 587	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ Lim 𝐴) → 𝑥 ∈ dom card)
63		cardid2 9913	. . . . . . . . 9 ⊢ (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
64	62, 63	syl 17	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ Lim 𝐴) → (card‘𝑥) ≈ 𝑥)
65	64	ensymd 8979	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ Lim 𝐴) → 𝑥 ≈ (card‘𝑥))
66	53, 54, 65	syl2anc 584	. . . . . 6 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (card‘𝑥))
67		simprr 772	. . . . . 6 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (cf‘𝐴) = (card‘𝑥))
68	66, 67	breqtrrd 5138	. . . . 5 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (cf‘𝐴))
69	51	simprd 495	. . . . 5 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → ∪ 𝑥 = 𝐴)
70	53, 68, 69	3jca 1128	. . . 4 ⊢ ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴))
71	70	ex 412	. . 3 ⊢ (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴)))
72	71	eximdv 1917	. 2 ⊢ (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴)))
73	49, 72	mpd 15	1 ⊢ (Lim 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708   ≠ wne 2926  ∃wrex 3054  {crab 3408  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  ∪ cuni 4874  ∩ cint 4913  ∩ ciin 4959   class class class wbr 5110  dom cdm 5641  Ord word 6334  Oncon0 6335  Lim wlim 6336  ‘cfv 6514   ≈ cen 8918  cardccrd 9895  cfccf 9897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-er 8674  df-en 8922  df-dom 8923  df-card 9899  df-cf 9901

This theorem is referenced by: (None)